The test function conjecture for parahoric local models

Let $G$ be a reductive group over $F$, let $\mathcal G$ be a parahoric group scheme, and let $E$ be the reflex field with maximal unramified subextension $E_0$. Let $M_{\{\mu\}}$ be the corresponding local model, $M_{\{\mu\},E}$ its generic fibre, and set

$$d_\mu=\dim M_{\{\mu\},E}.$$

Write $R\Psi_{M_{\{\mu\}}}(\operatorname{IC}_{\{\mu\}})$ for the nearby-cycles complex, let $\Phi_E$ denote geometric Frobenius, and let $z^{\rm ss}_{\{\mu\}}$ be the associated central Bernstein function. The semisimple Frobenius trace is viewed as an element of $\mathcal Z(G(E_0),\mathcal G(\mathcal O_{E_0}))$.

Test function conjecture. As elements of $\mathcal Z(G(E_0),\mathcal G(\mathcal O_{E_0}))$,

$$\operatorname{tr}^{\rm ss}\bigl(\Phi_E\mid R\Psi_{M_{\{\mu\}}}(\operatorname{IC}_{\{\mu\}})\bigr)=(-1)^{d_\mu}\cdot z^{\rm ss}_{\{\mu\}}.$$

This is the local-model formulation of the Test Function Conjecture for parahoric level, relating semisimple Frobenius traces on nearby cycles to central Bernstein functions. The supplied span does not state a resolution, so the conjecture is recorded as open.